SudokuSolver Forum

This is the sort of Assassin I really enjoy solving. Keep making progress with lots of nooks, crannies and killer stuff to explore but it keeps resisting! Found 3 ways to crack but there must be more. So, come on team, lets make this the widest Assassin ever! Does anyone remember a recent puzzle with more than 3 ways to crack it?

A big thankyou to Børge for the images. This is a killer X, so 1-9 cannot repeat on the diagonals.

Assassin 198


Cheers
Ed

Thanks Ed for a challenging Assassin; it's great to have you back posting puzzles again!

I really struggled with this puzzle so I suspect I didn't find any of Ed's 3 ways to solve it; there must be something important that I missed. When I later worked through Ed's walkthrough, further down this thread, I saw that I'd missed his step 11; possibly the most important thing I missed.


Here is my walkthrough for A198.

Prelims

a) R12C1 = {14/23}
b) R1C45 = {12}
c) R2C56 = {29/38/47/56}, no 1
d) 15(2) cage at R3C3 = {69/78}
e) R3C45 = {29/38/47/56}, no 1
f) 6(2) cage at R6C6 = {15/24}
g) R8C45 = {16/25/34}, no 7,8,9
h) R89C9 = {19/28/37/46}, no 5
i) R9C56 = {19/28/37/46}, no 5
j) 20(3) cage at R1C6 = {389/479/569/578}, no 1,2
k) 19(3) cage in N4 = {289/379/469/478/568}, no 1
l) 19(3) cage at R4C7 = {289/379/469/478/568}, no 1
m) 19(3) cage at R6C9 = {289/379/469/478/568}, no 1
n) 21(3) cage in N7 = {489/579/678}, no 1,2,3

1. Naked pair {12} in R1C45, locked for R1 and N2, clean-up: no 3,4 in R2C1, no 9 in R2C56, no 9 in R3C45

2. 45 rule on R12 2 innies R2C27 = 8 = {17/26/35}, no 4,8,9

3. 45 rule on R89 2 innies R8C38 = 14 = {59/68}

4. 45 rule on C1 2 outies R28C2 = 5 = [14/23/32], clean-up: no 1,2,3 in R2C7 (step 2)
4a. Min R2C7 = 5 -> max R3C78 = 7, no 7,8,9 in R3C78

5. 45 rule on C9 2 outies R28C8 = 14 = {59/68}

[Here I looked at interactions between R12C1, R2C2 and 6(2) cage at R6C6 along D\ but they are a bit “chainy” so I’ve left them for now.]

6. 45 rule on N3 1 outie R1C6 = 1 innie R3C9 + 5, R1C6 = {6789}, R3C9 = {1234}

[I saw the 45 on N7 next but it’s simpler to put the 45 on N8 before it.]

7. 45 rule on N8 2 innies R8C6 + R9C4 = 15 = {69/78}

8. 45 rule on N7 1 outie R9C4 = 1 innie R7C1 + 3, R7C1 = {3456}
8a. 45 rule on N7 3 innies R7C1 + R9C23 = 10 = {136/145/235} (cannot be {127} because R7C1 doesn’t contain 1,2,7), no 7,8,9
8b. 6 of {136} must be in R7C1 (R9C234 cannot be {16}6), no 6 in R9C23

9. 13(3) cage at R2C2 = {148/157/238/256} (cannot be {139/247/346} which clash with R12C1), no 9

10. 18(3) cage at R5C1 = {369/378/459/468/567} (cannot be {189/279} because R7C1 only contains 3,4,5,6), no 1,2

11. Max R12C1 + R2C2 = 8 so must contain 1, locked for R2 and N1
11a. 1 in N1 only in R2C12, CPE no 1 in R4C1

12. 45 rule on N14 4(3+1) outies R245C4 + R7C1 = 27
12a. Max R7C1 = 6 -> min R245C4 = 21, no 1,2,3 in R25C4

13. Killer quad 6,7,8,9 in 13(3) cage, R8C6 + R9C4 and R9C56, locked for N8, clean-up: no 1 in R8C45

14. Max R8C2 + R8C45 = 11 so must contain 2 (because there’s no 1), locked for R8, clean-up: no 8 in R9C9

15. Hidden killer pair 1,2 in R7C1 + R9C23 and 14(3) cage for N7, R7C1 + R9C23 contains one of 1,2 -> 14(3) cage must contain one of 1,2
15a. 14(3) cage = {149/239/248/257} (cannot be {158/167} because R8C2 only contains 2,3,4, cannot be {347/356} which don’t contain 1 or 2), no 6

[I ought to have spotted the next step sooner.]
16. 45 rule on C123 4 outies R2459C4 = 30 = {6789}, locked for C4, clean-up: no 3,4,5 in R3C5

17. 9 in N2 only in R13C6 + R2C4
17a. 45 rule on N2 3 innies R13C6 + R2C4 = 20 = {389/479/569}
17b. 3,4,5 only in R3C6 -> R3C6 = {345}

18. 9 in R3 only in R3C23, locked for N1, CPE no 9 in R45C3

19. 1 in N3 only in R3C789
19a. 12(3) cage at R2C7 = {147/156/237/246} (cannot be {345} which clashes with R3C46, ALS block)
19b. 45 rule on N3 3 innies R1C78 + R3C9 = 15 = {159/249/258/348} (cannot be {168/267} because 20(3) cage at R1C6 cannot be 6{68}/7{67}, cannot be {357/456} which clash with 12(3) cage at R2C7 when it contains 1), no 6,7
19c. 4 of {348} must be in R3C9 (because 20(3) cage at R1C6 cannot be 8{48}), no 3 in R3C9, no 8 in R1C6 (step 6)

20. 18(3) cage in N3 = {279/369/378/468} (cannot be {459} which clashes with R1C78 + R3C9, cannot be {567} which clashes with R2C7), no 5, clean-up: no 9 in R8C8 (step 5), no 5 in R8C3 (step 3)

21. 19(3) cage at R6C9 = {289/469/478/568} (cannot be {379} because R8C8 only contains 5,6,8), no 3

22. 5 in C9 only in 12(3) cage at R3C9 = {156/345} or in 19(3) cage at R4C7 = {568}
-> no 4,6 in R89C9 (locking-out cages)

23. 12(3) cage at R3C9 = {129/147/156/237/246/345} (cannot be {138} which clashes with R89C9), no 8

24. 19(3) cage at R6C9 cannot be {478}, here’s how
{478} => R89C9 = {19} => R6789C9 = {47}{19} clashes with 12(3) cage at R3C9
-> 19(3) cage at R6C9 (step 21) = {289/469/568}, no 7

25. 19(3) cage at R6C9 (step 24) = {289/469/568}
25a. 6 of {469} must be in R8C8 => R8C38 = [86] (step 3)
25b. -> 8 in 19(3) cage at R6C9 or 8 in R8C3, CPE no 8 in R8C9, clean-up: no 2 in R9C9

26. 18(3) cage in N3 (step 20) = {279/369/378/468}
26a. 9 of {279/369} must be in R2C8 (R12C9 cannot be {39} which clashes with R89C9), no 9 in R12C9

27. 19(3) cage at R6C9 cannot be {469}, here’s how
{469} => R67C9 = {49} => R89C9 = {37} => 12(3) cage at R3C9 must contain 5 = {156} => R12C9 cannot be [82] because 18(3) cage in N3 cannot be [828]
-> 19(3) cage at R6C9 (step 24) = {289/568}, no 4

28. 9 in R1 only in 20(3) cage at R1C6 = {389/479/569}
28a. R1C78 + R3C9 (step 19b) = {159/249/348} (cannot be {258} because 20(3) cage at R1C6 doesn’t contain both of 5,8)
28b. 12(3) cage at R2C7 (step 19a) = {156/237/246} (cannot be {147} which clashes with R1C78 + R3C9)
28c. 18(3) cage in N3 (step 20) = {279/378/468} (cannot be {369} which clashes with 12(3) cage)
28d. 6 of {468} must be in R2C89 (R1C9 = 6 clashes with 20(3) cage = 6{59}), no 6 in R1C9

29. 6 in R1 only in R1C23 + R1C6, CPE no 6 in R2C4

30. 25(4) cage at R1C2 cannot be {4678}, here’s how
{4678} => R1C1 = 3, R1C6 = 9 (hidden single in N2) => R3C9 = 4 (step 6) => R1C78 = {38} (step 28a) clashes with R1C1
30a. -> 25(4) cage at R1C2 = {2689/3589/3679/4579} -> R2C4 = 9, clean-up: no 6 in R3C3, no 3 in R3C6 (step 17a), no 4 in R3C9 (step 6), no 6 in R7C1 (step 8), no 6 in R8C6 (step 7), no 5 in R8C8 (step 5), no 9 in R8C3 (step 3)

31. Naked pair {68} in R28C8, locked for C8

32. Naked pair {68} in R8C38, locked for R8, clean-up: no 7 in R9C4 (step 7), no 4 in R7C1 (step 8)
32a. 7 in C4 only in R45C4, locked for N5, CPE no 7 in R4C23

33. Killer pair 6,8 in 15(2) cage at R3C3 and R8C8, locked for D\

34. 20(3) cage at R1C6 (step 28) = {479/569} (cannot be {389} because R1C6 only contains 6,7), no 3,8
34a. 18(3) cage in N3 (step 28c) = {378/468}, no 2
34b. 2 in N3 only in R3C789, locked for R3

35. 45 rule on C789 4 outies R1568C6 = 23
35a. Max R168C6 = 21 -> min R5C6 = 2

36. R7C1 + R9C23 (step 8a) = {145/235}, 5 locked for N7

37. 21(3) cage in N7 = {489/678}, 8 locked for N7

38. 14(3) cage in N7 (step 15a) = {149/239}, no 7, 9 locked for C1 and N7, clean-up: no 4 in R7C23 (step 37)
38a. 4 of {149} must be in R8C2 -> no 4 in R89C1
38b. 7 in N7 only in R7C23, locked for R7

39. Deleted

40. 7 in N8 only in R8C6 + R9C56, CPE no 7 in R9C78

41. 18(3) cage at R5C1 (step 10) = {378/567} (cannot be {468} because R7C1 only contains 3,5), no 4, 7 locked for C1 and N4

42. Naked pair {68} in R8C38, CPE no 8 in R3C3 using D\, clean-up: no 7 in R4C4
[I ought to have spotted this when I did step 32.]

43. R5C4 = 7 (hidden single in C4)
43a. R6C1 = 7 (hidden single in C1)
43b. 18(3) cage at R5C1 (step 41) = {378/567}
43c. 6,8 only in R5C1 -> R5C1 = {68}

44. 19(3) cage in N4 = {289/469} (cannot be {568} which clashes with R5C1), no 3,5, 9 locked for N4
44a. Killer pair 6,8 in R5C1 and 19(3) cage, locked for N4
44b. R35C1 = {68} (hidden pair in C1)

45. 25(4) cage at R1C2 (step 30a) = {3589/3679/4579} (cannot be {2689} which clashes with R3C1), no 2
45a. 2 in N1 only in R2C12, CPE no 2 in R4C1
45b. Killer pair 3,4 in R1C1 and 25(4) cage at R1C2, locked for N1

46. Naked triple {345} in R147C1, locked for C1
46a. 14(3) cage in N7 (step 38) = {149/239}
46b. 3,4 only in R8C2 -> R8C2 = {34}
[I originally did this step using hidden killer pair 1,2 for C1; then I saw the simpler naked triple.]

47. R5C4 = 7 -> 22(5) cage at R3C2 = {12379/12478/13567/23467}
47a. 6,8,9 only in R3C2 -> R3C2 = {689}

48. 5 in N1 only in 25(4) cage at R1C2 (step 45) = {3589/4579}, no 6
48a. 6 in N1 only in R3C12, locked for R3, clean-up: no 5 in R3C4

49. R1C6 = 6 (hidden single in R1) -> 20(3) cage at R1C6 (step 34) = {569} (only remaining combination), 5 locked for R1 and N3, clean-up: no 5 in R2C56, no 4 in R9C5
49a. R3C9 = 1 (step 6), clean-up: no 9 in R89C9
49b. R2C3 = 5 (hidden single in N1), R3C6 = 5 (hidden single in N2), clean-up: no 1 in R7C7
49c. R3C6 = 5 -> R4C56 = 10 = {19/28}/[64], no 3,4 in R4C5, no 3 in R4C6

50. Naked pair {37} in R89C9, locked for C9 and N9

51. 7 in R1 only in R1C23, locked for N1 -> R3C3 = 9, R4C4 = 6, both placed for D\, R8C8 = 8, R2C8 = 6, R2C7 = 7, R8C3 = 6, R9C4 = 8, R8C6 = 7 (step 7), R89C9 = [37], R8C2 = 4, clean-up: no 4 in R4C6 (step 49c), no 2,3 in R9C56

52. 14(3) cage in N7 (step 38) = {149} (only remaining combination), no 2, 1 locked for C1 and N7 -> R2C1 = 2, R1C1 = 3, R2C2 = 1, both placed for D\, R7C1 = 5, R4C1 = 4

53. 19(3) cage in N4 (step 44) = {289} (only remaining combination), no 6, 2,8 locked for N4 -> R5C1 = 6, R3C12 = [86], R3C5 = 7, R3C4 = 4

54. Naked pair {13} in R45C3, locked for C3 and N4 -> R4C2 = 5, R9C23 = [32], R6C3 = 8

55. Naked pair {25} in R8C45, locked for R8 and N8

56. R9C56 = [64] (cannot be {19} which clashes with R9C1), R6C6 = 2, R7C7 = 4, placed for D\, R5C5 = 5, clean-up: no 8 in R4C56 (step 49c)
56a. Naked pair {19} in R4C56, locked for R4 and N5 -> R6C4 = 3, placed for D/, R6C5 = 4, R45C3 = [31], R4C9 = 2, R5C9 = 9 (step 23), R67C9 = [56], R5C6 = 8, R5C7 = 3, R5C8 = 4, R6C78 = [61], R7C8 = 2 (cage sum)

57. R2C56 = [83], R7C456 = [139], R4C6 = 1, placed for D/, R9C1 = 9

and the rest is naked singles without using the diagonals.

Happy to have such a great assassin from you Ed ! It offers some amazing possibilities that were difficult for me to explore. Here is a partial walkthrough that certainly could be improved : please let me know.

PARTIAL WALKTHROUGH FOR ASSASSIN 198

This one turned out to be a real Assassin! Harder than anticipated. Thankyou Andrew and manu for your walk-throughs. Love manu's creativity!!

This is the way I solved it originally. Step 4 is the key though it has some tricky moves (steps 11, 21a and 27).

After this solution I had a look at the SS log (using ALT settings) and noticed how harmless it looks !! ( (see below - though gets a score around 1.60). So, tried again a few times. Think I finally found a solution that is technically easier. I've posted marks for where I went the different ways if anyone wants to try. I'll show the extra ways I found after the next Assassin is taken.

Partial Walkthrough for Assassin 198
Note: this is an optimized solution so some obvious eliminations are left out. However, I try and do clean-up as I go. Please let me know of any mistakes or corrections
Prelims
i. 5(2)n1 = {14/23}
ii. 3(2)n2 = {12}
iii. 20(3)n2: no 1,2
iv. 11(2)r2c5 & r3c4: no 1
v. 15(2)n1 = {69/78}
vi. 19(3)n6: no 1
vii. 19(3)n4: no 1
viii. 6(2)n5 = {15/24}
ix. 21(3)n7: no 1,2,3
x. 7(2)n8: no 7,8,9
xi. 10(2)n9: no 5
xii. 10(2)n8: no 5

1. naked pair 1,2 at r1c45: both locked for r1 and n2
1a. no 3,4 in r2c1
1b. no 9 in two 11(2)s in n2

2. "45" on c1: 2 outies r28c2 = 5 = {14/23}

3. "45" on c123: 4 outies r2459c4 = 30 = {6789}: all locked for c4
3a. no 3,4,5 in r3c5
3b. no 1 in r8c5

4. 25(4)n1 = {1789/2689/3589/3679/4579/4678}: must have one of 1,2,3,4 which is only in n1
4a. ->Killer quad 1,2,3,4 with 5(2)n1 and r2c2: all locked for n1
4b. 1 and 2 locked for r2

5. "45" on r12: 2 innies r2c27 = 8 = [17/26/35](no 4,8,9)
5a. r2c7 = (567)
5b. no 1 in r8c2 (h5(2)r28c2)

6. min. r2c7 = 5 -> max. r3c78 = 7 (no 7,8,9)

7. "45" on n3: 1 outie r1c6 - 5 = 1 innie r3c9
7a. r1c6 = (6..9), r3c9 = (1..4)

8. "45" on n2: 3 innies r1c6 + r2c4 + r3c6 = 20 and must have 9 for n2 = {389/479/569}
8a. has one of 3,4,5 which is only in r3c6 = (345)

9. 13(3)n1: {139/247/346} blocked by 5(2)n1 = [1/3, 2/4, 3/4..]
9a. = {148/157/238/256}(no 9) = [1/2..]

10. 9 in r3 only in n1: 9 locked for n1

11. "45" on n1: 2 outies r2c4 + r4c1 + 2 = 2 innies r3c23
11a. 2 innies must have 9 for n1 -> min. = [59] = 14
11b. -> min. 2 outies = 12 (no 1,2)
11c. 13(2)n1 must have 1/2 (step 9a) -> r2c2 = (12)
11d. no 5 in r2c7 (h8(2)r2c27)
11e. max. r3c78 = 6 (no 6)
11f. no 2 in r8c2 (h5(2)r28c2)

12. "45" on n8: 2 innies r8c6 + r9c4 = 15 = {69/78}

13. 13(3)n8 must have one of 6,7,8,9 to reach the cage sum (3+4+5 = 12)
13a. ->killer quad 6,7,8,9 with 10(2) and h15(2): 6 locked for n8
13b. no 1 in 7(2)n8

14. 7(2)n8: {34} blocked by r8c2
14a. = {25} only: both locked for n8 and r8
14b. no 8 in 10(2)n8
14c. no 8 in r9c9

15. "45" on r89: 2 innies r8c38 = 14 = {68} only: both locked for r8
15a. CPE from D\-> no 6,8 in r3c3
15b. no 7,9 in r4c4
15c. no 7,9 in r9c4 (h15(2)n8)
15d. no 2,4 in r9c9

16. naked pair 6,8 in r4c4 & r8c8: both locked for D\
16a. no 4 in r8c9

17. "45" on c9: 2 outies r28c8 = 14 = {68} only: both locked for c8

18. 10(2)n9 = {19/37} = [3/9..]
18a. -> 18(3)n3: {369} blocked since r12c9 must be must be {39}
18b. 18(3) must have 6 or 8 for r2c8 = {378/468/567}(no 9)

19. r2c4 = 9 (hsingle r2)
19a. -> no 4 in r3c9 (i/o n3 = -5)
19b. r5c4 = 7 (hsingle c4)

20. naked pair (12) in r2c12: Locked for n1

It's still not cracked so various ways to go from here.

Code:
.-------------------------------.-------------------------------.-------------------------------.
| 34        345678    345678    | 12        12        678       | 3456789   34579     345678    |
| 12        12        345678    | 9         345678    345678    | 67        68        345678    |
| 5678      5689      79        | 345        678      345       | 12345     12345     123       |
:-------------------------------+-------------------------------+-------------------------------:
| 345678    12345689  12345689  | 68        12345689  12345689  | 23456789  234579    123456789 |
| 12345689  2345689   12345689  | 7         123459    12345689  | 12345689  23459     12345689  |
| 123456789 23456789  23456789  | 12345     12345689  1245      | 123456789 1234579   23456789  |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 456789    456789    | 134       1346789   1346789   | 1245      1234579   23456789  |
| 13479     34        68        | 25        25        79        | 13479     68        13479     |
| 123456789 123456789 123456789 | 68        134679    134679    | 123456789 1234579   1379      |
'-------------------------------.-------------------------------.-------------------------------'

Original cracking

21. 25(4)n1 = 9{358/367/457} = [5/6..] but not both
21a. -> r3c12 must have 5/6 for n1 (no eliminations yet)

22. {56} blocked from 11(2)r3c4 by step 21a (no 5 or 6)

23. 6 in r3 only in n1: 6 locked for n1

24. 25(4)n1 = 9{358/457}
24a. 5 locked for n1

25. 13(3)n1 = {148/157/247/256} = [4/5..]
25a. -> r4c1 = (45)

26. "45" on n7: 1 outie r9c4 - 3 = r7c1
26a. r7c1 = (35)

The cracker
27. h5(2)r28c2 = [14/23] = [1/3..]
27a. -> combined half cage 5(2)n1 and r7c1 <> [413] since it clashes with h5(2)
27b. also can't be [415] since r4c1 = (45)
27c. -> r127c1 must be [325]
Much easier now.

Cheers
Ed

Now an ALT ending which is technically simpler but took the longest to find!
ALT ending
21. split cage 16(3) at r1c23 + r2c3
21a. ->"45" on r1: 1 outie r2c3 + 6 = 2 innies r1c19
21b. -> no 6 in r1c9 (IOU)

22. 20(3)r1c6 must have 9 for n3 = {389/479/569} = 6/7/8/ but not two of -> no 6,7,8 in r1c78

23. 6 in n3 only in r2: 6 locked for r2
23a. no 5 in 11(2)r2c5

24. 5 in n2 only in r3: 5 locked for r3

25. 5 in n1 only in split cage 16(3)r1c2 = {358/457}(no 6)

26. r1c6 = 6 (hsingle r1)

much easier now.

Very enjoyable puzzle.
Ed